Kinematics (DP IB Analysis & Approaches (AA): HL): Exam Questions

A cricket ball is projected vertically upwards from ground level. Its height above the ground, $h$ metres, $t$ seconds after it is projected can be modelled by $h\left(t\right)=13t-4.9t^2$, for $t\ge 0$.

Find the times at which the ball is 3 m above the ground.

Find the length of time for which the ball is at least 3 m above the ground.

A player catches the ball on its way down, when it is 0.8 m above the ground.

Find the length of time that the ball is in the air.

Find the total distance travelled by the ball before it is caught.

Find the velocity of the ball when $t=1$.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=2^t-2$, for $0\le t\le 4$.

Find the acceleration of the particle when $t=2$.

Find the value of $t$ when the particle is at rest.

Find the total distance travelled by the particle.

A particle moves in a straight line. Its acceleration, $a$ m s^-2, at time $t$ seconds is given by $a\left(t\right)=\frac{1}{t^2}+\sin t$, for $t\ge 1$.

When $t=1$, the velocity of the particle is 1 m s^-1.

Find an expression for the velocity, $v$ m s^-1, of the particle at time $t$.

Find the velocity of the particle when $t=2$.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=\sin t+\cos 2t$, for $0\le t\le 5$.

Find the values of $t$ when the particle is at rest.

Hence find the value of $t$ at which the particle changes direction.

Find the distance travelled by the particle in the first second of its motion.

Find the acceleration of the particle at the instant it changes direction.

Find the displacement of the particle from its starting point when $t=5$.

A marble is projected along a marble run that lies in a vertical plane through a fixed point $\mathrm{O}$. The height of the marble above $\mathrm{O}$, $h$ centimetres, $t$ seconds after it is projected can be modelled by $h\left(t\right)=15-10t+2t^2$, for $0\le t\le 6$.

Write down the initial height of the marble above $\mathrm{O}$.

Find the time at which the marble reaches its lowest point, and the total vertical distance travelled by the marble up to that time.

Find the velocity and the acceleration of the marble in the vertical direction when $t=6$.

A particle moves in a straight line. Its displacement, $s$ metres, from a fixed point $\mathrm{P}$ at time $t$ seconds is given by $s\left(t\right)=\frac{t^3}{20}-\frac{17t^2}{8}+18t-4$, for $0\le t\le 30$.

Write down the initial distance of the particle from $\mathrm{P}$.

Find the displacement of the particle from $\mathrm{P}$ when $t=30$.

Find an expression for the velocity, $v$ m s^-1, of the particle at time $t$.

Find the values of $t$ when the particle is at rest.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=t^3\cos t$, for $t\ge 0$.

Find an expression for the acceleration, $a$ m s^-2, of the particle at time $t$.

Hence find the acceleration of the particle when $t=5.2$.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=\sqrt{t^2+2t}-3$, for $0\le t\le 5$.

Find the acceleration of the particle when $t=3.7$.

(i) Find the displacement of the particle from its starting position when $t=1$.

(ii) Explain what your answer to part (b)(i) tells you about the motion of the particle for $0\le t\le 1$.

Find the total distance travelled by the particle.

A particle P moves in a straight line. Its velocity, $v$ m s^-1, after $t$ seconds is given by

$v\left(t\right)=\frac{1}{2}\sin t-3t^{\cos t}+8$, for $0\le t\le 10$.

The following diagram shows the graph of $v$.

Find the initial velocity of P.

Find the maximum speed of P.

Write down the number of times that the acceleration of P is 0 m s^-2.

Find the acceleration of P when it first changes direction.

A ball is placed at the midpoint, X, of a horizontal tube of length 120 cm. A player at each end of the tube pumps air into it to move the ball.

The following graph shows the velocity, $v$ cm s^-1, of the ball at time $t$ seconds during a 20-second game.

Write down the maximum speed of the ball and the interval of time during which it occurs.

There are two intervals of time during which the ball is decelerating. Find the magnitude of the deceleration of the ball after it first changes direction.

Find the total distance travelled by the ball during the game.

A particle moves in a straight line. Its displacement, $s$ metres, from a fixed point $\mathrm{O}$ at time $t$ seconds is given by $s\left(t\right)=\frac{1}{2}t-\sin 2t$, for $0\le t\le 6$.

Write down the number of times the particle changes direction.

Find an expression for the velocity, $v$ m s^-1, of the particle at time $t$.

Find the maximum velocity of the particle and the values of $t$ at which it occurs.

Find the percentage of the total distance travelled by the particle that is travelled in the first 2 seconds.

A particle P moves in a straight line. Its velocity, $v_{\mathrm{P}}$ m s^-1, at time $t$ seconds is given by

$v_{\mathrm{P}}\left(t\right)=t^2\cos \left(\frac{\pi }{4}t\right)$

for $0\le t\le 10$.

Write down the value of $t$ at which P first changes direction.

Find the total distance travelled by P while its speed is increasing.

A second particle, Q, also moves in a straight line. Its velocity, $v_{\mathrm{Q}}$ m s^-1, at time $t$ seconds is given by

$v_{\mathrm{Q}}\left(t\right)=6t+2$

for $0\le t\le 8$.

After $k$ seconds, the total distance travelled by Q is equal to the distance travelled by P while its speed is increasing.

Find the value of $k$.

The height, $h$ metres, of the water level in an estuary relative to the mean sea level, $t$ hours after midnight, can be modelled by

$h\left(t\right)=\sin \left(0.262t-0.5\right)-3\cos \left(0.524t\right)+1$

for $0\le t\le 24$.

Find the rate of change of the height of the water level at 7 am.

Find the percentage of the 24-hour period for which the water level is above the mean sea level.

A buoy is anchored in the estuary so that its horizontal position is fixed, but it moves up and down with the water level.

Find the total vertical distance travelled by the buoy during the 24-hour period.

A particle moves in a straight line, starting from a fixed point $\mathrm{P}$. Its velocity, $v$ m s^-1, at time $t$ seconds is given by

$v\left(t\right)=\{\begin{array}{cc}9t-3t^2,& 0\le t\le 4\\ -3t+\frac{16}{t^3}-\frac{1}{4},& 4<t\le 10\end{array}$

Find the maximum velocity of the particle and the value of $t$ at which it occurs.

Find an expression for the displacement, $s$ metres, of the particle from $\mathrm{P}$ at time $t$, given that $s\left(10\right)=-119.08$.

Find the total distance travelled by the particle.

A toy car moves along a straight horizontal track, starting from a fixed point $\mathrm{P}$. Its displacement, $s$ centimetres, from $\mathrm{P}$ at time $t$ seconds can be modelled by

$s\left(t\right)=\frac{1}{1000}t\left(t^2-190t+8400\right)$

for $0\le t\le 140$.

Sketch the graph of $s$ against $t$.

Find an expression for the acceleration, $a$ cm s^-2, of the toy car at time $t$. Hence find the acceleration of the toy car when $t=85$.

Find the greatest speed of the toy car while its displacement is decreasing.

A particle starts from a fixed point $\mathrm{X}$ and moves in a straight line. The following graph shows its velocity, $v$ m s^-1, at time $t$ seconds, for $0\le t\le 8$.

The particle is instantaneously at rest when $t=0$, $t=3$, $t=5$ and $t=8$.

Let $s\left(t\right)$ metres be the displacement of the particle from $\mathrm{X}$ at time $t$.

The particle travels 22 metres in the first 3 seconds. Also, $s\left(3\right)=s\left(7\right)$ and ${\int }_3^{5}v\mathrm{d}t=9$.

Write down the value of $s\left(5\right)-s\left(3\right)$.

Find the total distance travelled by the particle in the first 7 seconds.

A particle A moves along a horizontal straight line $L_1$. Its displacement, $s_{\mathrm{A}}$ metres, from a fixed point $\mathrm{P}$ on $L_1$ at time $t$ seconds is given by

$s_{\mathrm{A}}\left(t\right)=\frac{1}{2}t-2t^3{\mathrm{e}}^{-0.3t}+24$

for $0\le t\le 22$.

At the same time, a second particle, B, starts to move along a horizontal straight line $L_2$, parallel to $L_1$. Its velocity, $v_{\mathrm{B}}$ m s^-1, at time $t$ seconds is given by

$v_{\mathrm{B}}\left(t\right)=\frac{1}{2}t+12$

for $0\le t\le 22$.

Find the values of $t$ for which particle A is at $\mathrm{P}$.

Find the value of $t$ at which particle A first changes direction.

Find the total distance travelled by particle A in the first 8 seconds of its motion.

The displacement, $s_{\mathrm{B}}$ metres, of particle B is measured from a fixed point $\mathrm{Q}$ on $L_2$. It is given that $s_{\mathrm{A}}\left(0\right)=s_{\mathrm{B}}\left(5\right)$.

(i) Find an expression for $s_{\mathrm{B}}\left(t\right)$.

(ii) Find the displacement of each particle at the time when $s_{\mathrm{A}}\left(t\right)=s_{\mathrm{B}}\left(t\right)$.

Two particles, ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$, move along the same straight line. Their displacements, $s_1$ metres and $s_2$ metres, from a fixed point $\mathrm{O}$ at time $t$ seconds are given by

$s_1\left(t\right)=\frac{1}{2}\sin \left(t-0.9\right)-\cos \left(2t-1.8\right)-1$

$s_2\left(t\right)=\cos \left(6t-5.4\right)-\sin \left(t-0.9\right)+2.5$

for $0\le t\le 3$.

Find an expression for the distance between the two particles at time $t$.

Hence find

(i) the maximum distance between the particles

(ii) the value of $t$ at which this maximum distance occurs.

The particles collide once during this time.

Find the velocity of each particle 0.5 seconds before they collide.